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Mirrors > Home > MPE Home > Th. List > Mathboxes > submateqlem2 | Structured version Visualization version GIF version |
Description: Lemma for submateq 33333. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
Ref | Expression |
---|---|
submateqlem2.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
submateqlem2.k | ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) |
submateqlem2.m | ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) |
submateqlem2.1 | ⊢ (𝜑 → 𝑀 < 𝐾) |
Ref | Expression |
---|---|
submateqlem2 | ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz1ssnn 13550 | . . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
2 | submateqlem2.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
3 | 1, 2 | sselid 3976 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
4 | 3 | nnge1d 12276 | . . . 4 ⊢ (𝜑 → 1 ≤ 𝑀) |
5 | submateqlem2.1 | . . . 4 ⊢ (𝜑 → 𝑀 < 𝐾) | |
6 | 4, 5 | jca 511 | . . 3 ⊢ (𝜑 → (1 ≤ 𝑀 ∧ 𝑀 < 𝐾)) |
7 | 2 | elfzelzd 13520 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
8 | 1zzd 12609 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
9 | submateqlem2.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
10 | 9 | elfzelzd 13520 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
11 | elfzo 13652 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) | |
12 | 7, 8, 10, 11 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) |
13 | 6, 12 | mpbird 257 | . 2 ⊢ (𝜑 → 𝑀 ∈ (1..^𝐾)) |
14 | 2 | orcd 872 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁)) |
15 | submateqlem2.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
16 | nnuz 12881 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | eleqtrdi 2838 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1)) |
18 | fzm1 13599 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) |
20 | 14, 19 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
21 | 3 | nnred 12243 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
22 | 21, 5 | ltned 11366 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐾) |
23 | nelsn 4664 | . . . 4 ⊢ (𝑀 ≠ 𝐾 → ¬ 𝑀 ∈ {𝐾}) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝑀 ∈ {𝐾}) |
25 | 20, 24 | eldifd 3955 | . 2 ⊢ (𝜑 → 𝑀 ∈ ((1...𝑁) ∖ {𝐾})) |
26 | 13, 25 | jca 511 | 1 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∖ cdif 3941 {csn 4624 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 1c1 11125 < clt 11264 ≤ cle 11265 − cmin 11460 ℕcn 12228 ℤcz 12574 ℤ≥cuz 12838 ...cfz 13502 ..^cfzo 13645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-fzo 13646 |
This theorem is referenced by: submateq 33333 |
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